lsc.amss.ac.cnlsc.amss.ac.cn/~bzguo/papers/Nielf.pdf · 2013-10-07 · 1698 L.-F. Nie et al. (1927) in the late 1920s. In this model, the total population N is divided into sus- ceptible

1 23

Bulletin of Mathematical BiologyA Journal Devoted to Research at theJunction of Computational, Theoreticaland Experimental Biology OfficialJournal of The Society for MathematicalBiology ISSN 0092-8240Volume 75Number 10 Bull Math Biol (2013) 75:1697-1715DOI 10.1007/s11538-013-9865-y

A State Dependent Pulse Control Strategyfor a SIRS Epidemic System

Lin-Fei Nie, Zhi-Dong Teng & Bao-ZhuGuo

1 23

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Bull Math Biol (2013) 75:1697–1715DOI 10.1007/s11538-013-9865-y

O R I G I NA L A RT I C L E

A State Dependent Pulse Control Strategy for a SIRSEpidemic System

Lin-Fei Nie · Zhi-Dong Teng · Bao-Zhu Guo

Received: 19 December 2012 / Accepted: 30 May 2013 / Published online: 28 June 2013© Society for Mathematical Biology 2013

Abstract With the consideration of mechanism of prevention and control for thespread of infectious diseases, we propose, in this paper, a state dependent pulse vac-cination and medication control strategy for a SIRS type epidemic dynamic system.The sufficient conditions on the existence and orbital stability of positive order-1 ororder-2 periodic solution are presented. Numerical simulations are carried out to il-lustrate the main results and compare numerically the state dependent vaccinationstrategy and the fixed time pulse vaccination strategy.

Keywords SIRS epidemic model · State dependent pulse · Order-k periodicsolution · Orbital stability

1 Introduction

The control and hence eradication of infectious disease is one of the major concernsin the study of mathematical epidemiology. In the last several decades, the epidemicdynamical models which are described by the differential equations have played acrucial role in control and eradication of the infectious diseases. Perhaps an earli-est classical epidemic dynamical model was developed by Kermack and Mckendrick

L.-F. Nie (�) · Z.-D. TengCollege of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P.R. Chinae-mail: [email protected]

B.-Z. GuoAcademy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China

B.-Z. GuoSchool of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050,Johannesburg, South Africa

Author's personal copy

mailto:[email protected]

1698 L.-F. Nie et al.

(1927) in the late 1920s. In this model, the total population N is divided into sus-ceptible individuals, infected individuals, and recovered individuals, which are rep-resented shorthand by S, I , and R respectively. Many kinds of epidemic dynamicalmodels have been developed subsequently to analyze the spread and control of infec-tious diseases. We refer some of them to (Anderson and May 1979; Capasso 1993;Diekmann and Heesterbeek 2000; Ruan and Wang 2003) and the references therein.Based on these dynamical models, many researchers studied the evolution of an infec-tious disease from different perspectives. These include the existence of the thresholdvalue which is the index for the evolution and extinction of an infectious disease; thelocal or global stability of the disease-free equilibrium and endemic equilibrium; theexistence of periodic solutions; and the persistence and extinction of the disease,name just a few.

In the past years, with the progress of the globalization, the control of infec-tious diseases has been concerned increasingly interdisciplinary in different con-texts. Of many strategies, the efficient ways for elimination or control of infec-tious diseases are still immunization and medication (Sabin 1991; Ramsay et al.1994). One of the medications is the pulse vaccination and medication, which hasbecome a major topic in mathematical biology and mathematical epidemiology(see, e.g., Gao et al. 2006, 2007; Terry 2010; Wang et al. 2010; Liu et al. 2009;Pei et al. 2009, and the references therein). Using a SIR epidemic model, Shulginet al. (1998) showed that under a planned pulse vaccination region, the sate of systemconverges to a stable sate with which the size of infectious population is zero. Thisshows that the pulse vaccination may lead to the eradication of infectious disease pro-vided that the magnitude of vaccination keeps a rational proportion and the period ofpulses is sustained. D’Onofrio (2002) proposed a SEIR epidemic model based pulsevaccination strategy by which the local and global asymptotic stabilities of the peri-odic eradication solution are analyzed. Gao et al. (2011) proposed an impulsive SIRSepidemic model with periodic saturation incidence and vertical transmission. The ef-fects of periodic varying contact rate and mixed vaccination strategy on eradicationof infectious disease are studied. In all these studies, the impulses are supposed in afixed time interval, for which we may call the fixed-time pulse vaccination and med-ication strategy (FTPS for short). The theoretical results and statistical data show,however, FTPS is very different from the conventional strategies in leading to disease(viral infections, such as rabies, yellow fever, poliovirus, hepatitis B, and encephali-tis B) eradication at relatively low values of vaccination and medication (Agur et al.1983).

A noticed fact is that the transmission of different infectious diseases has differentways. For instance, measles, cholera and influenza, HIV/AIDS, pulmonary tuberculo-sis, hepatitis E virus and bacillary dysentery are transmitted irregularly, or transmittedthrough particular mechanism. The practice shows that these diseases cannot be erad-icated in a short time. A natural idea is therefore to keep the density of infections ata low level to avoid the spread of the disease. In this regard, exploring an effectiveand easily implemented control measure to keep the level of spread of diseases issignificant both theoretically and practically.

Recently, the state dependent impulsive feedback control strategy is appliedwidely to the control of spread of infectious disease due to its economic, high ef-


A State Dependent Pulse Control Strategy for a SIRS Epidemic 1699

ficiency, and feasibility nature. This idea can be found in many other areas like agri-cultural production and fishery industry where the control measures (such as catch-ing, poisoning, releasing the natural enemy, and harvesting) are taken only when thenumber of population reaches a threshold value. In this regard, it is different from thefixed-time pulse control strategy. Many works have been focused on the analysis ofmathematical models described by ordinary differential equations with state depen-dent pulse effects. In (Nie et al. 2010, 2009; Jiang and Lu 2006, 2007; Tang et al.2005), the dynamic behaviors of predator–prey models with state dependent pulseeffects are considered and the existence and stability of positive periodic solution bythe Poincaré map, properties of the Lambert W function, analogue of Poincaré crite-rion are obtained. In addition, Ross (1911) proposed a mathematical model to studythe spread between human beings and mosquitoes for malaria in earlier 1911, wherea concept of threshold density is introduced and it is concluded that “. . . in order tocounteract malaria anywhere we need not banish Anopheles there entirely—we needonly to reduce their numbers below a certain figure.”

Motivated by these facts, we propose, in this paper, a state dependent pulse vac-cination and medication control for a SIRS epidemic system. The difference of thedynamic behaviors between our model and the fixed-time pulse effect model is illus-trated in Sect. 2. Some sufficient conditions are presented in Sect. 3 for the existenceand stability of positive periodic solutions. The numerical simulations are carriedout in Sect. 4 for illustration. Through the numerical comparison with the fixed-timepulse control strategy, it is shown that the state dependent pulse vaccination and med-ication strategy is more effective and easily implemented to prevent the spread levelof the disease. Some concluding remarks are presented in Sect. 5.

2 Model Formulation and Preliminaries

A traditional autonomous SIRS epidemic model is of the following form:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

dS(t)

dt= b

(S(t) + I (t) + R(t)

) − βS(t)I (t) − bS(t) + αR(t),

dI (t)

dt= βS(t)I (t) − bI (t) − γ I (t),

dR(t)

dt= γ I (t) − bR(t) − αR(t),

(1)

where S(t), I (t), and R(t) stand for the numbers of susceptible, infected, and recov-ered individuals to the disease of a population at time t respectively, b > 0 representsthe immigration rate with the assumption that all newborns are susceptible, whichalso represents the death rate of susceptible, infected, and recovered groups, respec-tively. It is assumed that all susceptible group becomes infected at a rate βI , whereβ > 0 is the contact rate; the infected group becomes recovered who received life-time immunity at a constant rate γ > 0. Therefore, 1/γ is the mean infectious period.In addition, the recovered group becomes susceptible group again at a constant rateα > 0, so 1/α is the average immune time.



The system (1) has been studied extensively, and some of them can be found inAnderson and May (1991), Hethcote (1989) and the references therein. The succeed-ing results on system (1) are available in Ma et al. (2004).

Theorem 1

(i) If parameter R0 = β/(b + γ ) < 1, then system (1) admits only a globally asymp-totically stable disease-free equilibrium (1,0,0).

(ii) If parameter R0 > 1, then system (1) admits an unstable disease-free equilib-rium (1,0,0) and a unique globally asymptotically stable endemic equilibrium(S∗, I ∗,R∗), where

S∗ = b + γ

β, I ∗ = (b + α)(β − b − γ )

β(b + α + γ ), R∗ = γ (β − b − γ )

β(b + α + γ ).

Generally speaking, the immunization and medication control for infectious dis-ease can be modeled in three different forms: the continuous time control where thecontrol measure is taken once the disease is discovered; the fixed-time pulse controlaforementioned; and the state dependent pulse control presented in this paper. Amongthem the continuous time control is not practically implementable. Different with theusual continuous control or fixed-time pulse control strategy, we propose a state de-pendent pulse feedback control strategy. In addition, since the medication for someinfectious diseases is relatively short, we suppose that the procedure of medicationtakes pulse effect when the number of group I reaches the threshold value. We firstintroduce the following assumption (A) before building a new mathematical model.

(A) When the number of the infected individuals reaches the hazardous thresholdvalue H where, 0 < H < 1 − S∗ at time ti (H) at the ith time, the vaccinationand medication are taken and the numbers of susceptible, infected, and recoveredindividuals turn very suddenly to a great degree to (1 − m)S(t+i (H)), ((1 −p)I (t+i (H))), and R(t+i (H)) + mS(t+i (H)) + pI (t+i (H)), respectively, wherem,p ∈ (0, 1) are the vaccination intensity and medication intensification effort,respectively.

Under assumption (A), we propose a new control model of infectious diseasewhich is described by the following ordinary differential equation with the state de-pendent pulse effects:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dS(t)

dt= b

(S(t) + I (t) + R(t)

) − βS(t)I (t) − bS(t) + αR(t),

dI (t)

dt= βS(t)I (t) − bI (t) − γ I (t),

dR(t)

dt= γ I (t) − bR(t) − αR(t),

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

I < H,

�S(t) = S(t+

) − S(t) = −mS(t),

�I (t) = I(t+

) − I (t) = −pH,

�R(t) = R(t+

) − R(t) = mS(t) + pH

⎫⎪⎪⎬

⎪⎪⎭

I = H.

(2)



Remark 1 We point out that a priori time of pulse vaccination and medication is notassumed because it is taken at the time when the number of group I reaches thethreshold value H . So the pulse vaccination and medication time depend obviouslyon the solution, which makes our model “state dependent.”

Remark 2 The control parameters m, p, and hazardous threshold H rely heavily onthe characteristics of the disease. The choices of these values are very important fordifferent diseases, which are closely related to the spread or prevention of the disease.

Let R = (−∞, ∞) and R3+ = {(x, y, z)|x > 0, y > 0, z > 0}. The global existence

and uniqueness of solution for system (2) are guaranteed by the smoothness of theright-hand sides of system (2). For more details, we refer to Lakshmikantham et al.(1989).

Lemma 1 Suppose that (S, I,R) is a solution of system (2) with the initial value(S(t0), I (t0),R(t0)) ∈ R

3+. Then S(t) > 0, I (t) > 0, and R(t) > 0 for all t ≥ t0.

Proof For any initial value (S(t0), I (t0),R(t0)) ∈ R3+, we discuss all possible cases

by the relation of the solution (S, I,R) to the line L : I = H as follows.Case 1: The solution (S, I,R) intersects with L finitely many times.In this case, since the endemic equilibrium (S∗, I ∗,R∗) is globally asymptotically

stable, S(t) > 0, I (t) > 0, and R(t) > 0 for all t ≥ t0.Case 2: The solution (S, I,R) intersects with L : I = H infinitely many times.In this case, suppose that solution (S, I,R) intersects with L : I = H at times

tk, k = 1,2, . . . , limk→∞ tk = ∞. If the conclusion of Lemma 1 is invalid, then thereis a t∗ > t0 such that min{S(t∗), I (t∗),R(t∗)} = 0, and S(t) > 0, I (t) > 0, R(t) > 0for all t0 ≤ t < t∗. For this t∗, there is a positive integer n such that tn−1 ≤ t∗ < tn.There are three possible cases.

(i) I (t∗) = 0, S(t∗) ≥ 0, and R(t∗) ≥ 0. For this case, it follows from the secondand fifth equations of system (2) that

I(t∗

) = (1 − p)n−1I(t+0

)e

∫ t∗t0

(βS(τ)−b−γ )dτ> 0,

which leads to a contradiction with I (t∗) = 0.(ii) R(t∗) = 0, S(t∗) ≥ 0, and I (t∗) ≥ 0. It follows from the third and sixth equa-

tions of system (2) that

R(t∗

) ≥ R(t+0

)e− ∫ t∗

t0(b+γ )dt

+ pH(e− ∫ t∗

t1(b+γ )dt + e

− ∫ t∗t2

(b+γ )dt + · · · + e− ∫ t∗

tn−1(b+γ )dt )

> 0,

which leads to a contradiction with R(t∗) = 0.(iii) S(t∗) = 0, I (t∗) ≥ 0, and R(t∗) ≥ 0. It then follows from the first and fourth

equations of system (2) that

S(t∗

) ≥ (1 − m)n−1S(t+0

)e− ∫ t∗

t0(βI (τ)+b)dτ

> 0,



Fig. 1 The dynamic behavior of system (3) without pulse vaccination and medication (a), and with statedependent pulse vaccination and medication (b) (Color figure online)

which leads to a contradiction with S(t∗) = 0.

These contradictions show that such a t∗ does not exist. Therefore, S(t) > 0,I (t) > 0, and R(t) > 0 for all t ≥ t0. This completes the proof. �

Since from (2), the total population is normalized to unity that

S(t) + I (t) + R(t) = 1,

therefore system (2) is equivalent to⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

dS(t)

dt= b + α − βS(t)I (t) − (b + α)S(t) − αI (t),

dI (t)

dt= βS(t)I (t) − bI (t) − γ I (t),

⎫⎪⎪⎬

⎪⎪⎭

I < H,

�S(t) = S(t+

) − S(t) = −mS(t),

�I (t) = I(t+

) − I (t) = −pH,

}

I = H.

(3)

We assume, throughout this paper, that R0 = β/(b+γ ) > 1. That is to say, system(3) without pulse effects has a unique globally asymptotically stable endemic equi-librium E∗(S∗, I ∗) (see Fig. 1(a)). Based on the biological background of system (3)we only consider system (3) in the region R

2+ = {(S, I ) : S > 0, I > 0} where thebiology makes sense.

Let Γ be an arbitrary set in R2 and Y be an arbitrary point in R

2. The distancebetween Y and Γ is defined by d(Y,Γ ) = infY0∈Γ |Y − Y0|. Let X = (S, I ) be anysolution of system (3). We define the positive orbit starting from initial point X0 ∈ R

2+for t ≥ t0 as

O+(X0, t0) = {X(t) : X(t) = (

S(t), I (t)) ∈ R

2+, t ≥ t0,X(t0) = X0}.

Definition 1 (Orbital Stability, Hale and Kocak 1991) The trajectory O+(X0, t0) issaid to be orbitally stable if for any given ε > 0, there exists a constant δ = δ(ε) > 0



such that for any other solution X∗ of system (3), d(X∗(t),O+(X0, t0)) < ε for allt > t0 when d(X∗(t0),O+(X0, t0)) < δ.

Definition 2 (Orbital Asymptotical Stability, Hale and Kocak 1991) The trajectoryO+(X0, t0) is said to be orbitally asymptotically stable if it is orbitally stable, andthere exists a constant η > 0 such that for any other solution X∗ of system (3),limt→∞ d(X∗(t),O+(X0, t0)) = 0 when d(X∗(t0),O+(X0, t0)) < η.

To discuss the dynamic behavior of system (3), we define two sections to the vectorfield of system (3) by

Γp := {(S, I ) : 0 < S < 1 − (1 − p)H, I = (1 − p)H

}

and

ΓH := {(S, I ) : 0 < S < 1 − H,I = H

}.

For any point Pn(Sn,H) ∈ ΓH , suppose that the trajectory O+(Pn, tn) startingfrom the initial point Pn intersects section ΓH infinitely many times. That is, tra-jectory O+(Pn, tn) jumps to point P +

n ((1 − m)Sn, (1 − p)H) on section Γp dueto impulsive effects �S(t) = −mS(t) and �I (t) = −pH . Furthermore, trajectoryO+(Pn, tn) intersects section ΓH at point Pn+1(Sn+1,H), and then jumps to pointP +

n+1((1 − m)Sn+1, (1 − p)H) on section Γp again (see Fig. 1(b)). Repeating thisprocess, we have two impulsive point sequences {P +

n ((1 − m)Sn, (1 − p)H)} and{Pn(Sn,H)}, where Sn+1 is determined by Sn, m, p, H , and n = 1,2, . . . . We thendefine the Poincaré map of section ΓH :

Sn+1 = F(Sn,m,p,H). (4)

Definition 3 A trajectory O+(Pn, tn) of system (3) is said to be order-k periodic ifthere exists a positive integer k ≥ 1 such that k is the smallest integer for Sn+k = Sn.

Next, we consider the autonomous system with pulse effects of the following:⎧⎨

⎩

dx

dt= f (x, y),

dy

dt= g(x, y), ϕ(x, y) �= 0,

�x = ξ(x, y), �y = η(x, y), ϕ(x, y) = 0,

(5)

where f and g are continuous differentiable functions defined on R2 and ϕ is a suffi-

ciently smooth function with ∇ϕ �= 0. Let (μ, ν) be a positive T -periodic solution ofsystem (5). The following result comes from Corollary 2 of Theorem 1 of Simeonovand Bainov (1988).

Lemma 2 (Analogue of Poincaré Criterion) If the Floquet multiplier μ satisfies|μ| < 1, where

μ =n∏

j=1

κj exp

{∫ T

0

[∂f (μ(t), ν(t))

∂x+ ∂g(μ(t), ν(t))

∂y

]

dt

}



with

κj = (∂η∂y

∂ϕ∂x

− ∂η∂x

∂ϕ∂y

+ ∂ϕ∂x

)f+ + (∂ξ∂x

∂ϕ∂y

− ∂ξ∂y

∂ϕ∂x

+ ∂ϕ∂y

)g+∂ϕ∂x

f + ∂ϕ∂y

g

and f , g, ∂ξ/∂x, ∂ξ/∂y, ∂η/∂x, ∂η/∂y, ∂ϕ/∂x, and ∂ϕ/∂y have been calculatedat the point (μ(τj ), ν(τj )), f+ = f (μ(τ+

j ), ν(τ+j )), g+ = g(μ(τ+

j ), ν(τ+j )), and τj

(j ∈ N) is the time of the j th jump, then (μ, ν) is orbitally asymptotically stable.

3 Main Results

Since endemic equilibrium E∗(S∗, I ∗) is globally asymptotically stable, any pos-itive solution of system (3) without state dependent pulse will eventually tend toE∗(S∗, I ∗). From the geometrical structure of the phase space of system (3), it iseasily to obtain that any trajectory of (3) without state dependent pulse starting fromdomain I := {(S, I ) ∈ R

2 | S′ < 0, I ′ < 0} will enter into domains II := {(S, I ) ∈R

2 | S′ > 0, I ′ < 0}, III := {(S, I ) ∈ R2 | S′ > 0, I ′ > 0}, or IV := {(S, I ) ∈ R

2 | S′ <0, I ′ > 0} by order, and eventually tend to E∗(S∗, I ∗) (see Fig. 1(a)). Therefore, ifH ≤ I ∗ = (b + α)(β − b − γ )/β(b + α + γ ), then the trajectory with given initialvalue (S0, I0), I0 = (1 − p)H intersects section ΓH infinitely many times. However,if H > I ∗, the trajectory starting from initial point (S0, I0) with I0 = (1 − p)H mayintersect section ΓH finitely many times. In this section, we give some sufficient con-ditions for the existence and stability of positive periodic solutions in the cases ofH ≤ I ∗ and H > I ∗, respectively.

3.1 The Case of H ≤ I ∗

The first result is on the existence of positive order-1 periodic solution.

Theorem 2 For any m,p ∈ (0, 1), system (3) admits a positive order-1 periodicsolution.

Proof Let point E1(ε, (1 − p)H) ∈ Γp for sufficiently small ε with ε ≤ (1 − m)S∗.In view of the geometrical structure of the phase space of system (3), the trajectoryO+(E1, t0) of system (3) starting from the initial point E1 in domain II will enterinto domains III or IV and intersect section Γp at point E1p(ε1p, (1 − p)H), andthen intersect section ΓH at point F1(S1,H), where S∗ < S1. At point F1, the tra-jectory O+(E1, t0) jumps to the point E2((1 − m)S1, (1 − p)H) on section Γp dueto impulsive effects �S(t) = −mS(t) and �I (t) = −pH . Furthermore, trajectoryO+(E1, t0) intersects section ΓH at point F2(S2,H).

Since ε ≤ (1 −m)S∗, E1 is on the left of E2. We claim that F2 is on the left of F1.In fact, if F2 is on the right of F1, then the orbits E1F1 and E2F2 intersect at somepoint D(S0, I0). This shows that there are two different solutions which start from thesame initial point D(S0, I0). This contradicts the uniqueness of solutions for system(3). Therefore, it follows from (4) that S2 = F(S1,m,p,H) and

F(S1,m,p,H) − S1 = S2 − S1 < 0. (6)



On the other hand, suppose that the curve L : βSI −bI −γ I = 0 intersects sectionΓp at point A0((b + γ )/β, (1 − p)H). Then the trajectory O+(A0, t0) starting fromthe initial point A0 intersects section ΓH at point B1(S1,H) and then jumps to pointA1((1 − m)S1, (1 − p)H) on section Γp and finally reaches point B2(S2,H) in sec-tion ΓH again. If there is a positive constant m∗ such that (1 − m∗)S1 = (b + γ )/β ,then A1 coincides with A0 for m = m∗ ∈ (0,1), that is, B1 coincides with B2. Other-wise, A1 is on the left of A0 for (1 − m)S1 < (b + γ )/β and A1 is on the right of A0for (1 − m)S1 > (b + γ )/β . However, from the geometrical structure of phase spaceof system (3), B2 is on the right of B1 for any m ∈ (0,m∗) ∪ (m∗,1) (see Fig. 1(b)).

To sum up, we get, from the above discussions, that

(i) When S1 = S2, system (3) has positive order-1 periodic solution.(ii) When S1 < S2,

F(S1,m,p,H) − S1 = S2 − S1 > 0. (7)

By (6) and (7), it follows that the Poincaré map (4) has a fixed point. This amountsto saying that system (3) has a positive order-1 periodic solution. The proof is com-plete. �

Remark 3 From Theorem 2, we see that when H ≤ I ∗, system (3) always admits apositive order-1 periodic solution. In addition, from the geometrical structure of thephase space of system (3), it is easily to get that the disease could be controlled belowthe threshold value H due to the state dependent pulse strategy.

The next result is on the orbital stability of positive order-1 periodic solution ofmodel (3).

Theorem 3 Let (φ,ψ) be a positive order-1 periodic solution of model (3) withperiod T . If

|μ| = |κ| exp

{

−∫ T

0

[βψ(t) + b + α

]dt

}

< 1, (8)

where

κ = (1 − m)[β(1 − m)φ(T ) − b − γ ]βφ(T ) − b − γ

,

then (φ,ψ) is orbitally asymptotically stable.

Proof Suppose that (φ,ψ) intersects the sections Γp and ΓH at points E+((1 −m)φ(T ), (1 − p)H) and E(φ(T ),H), respectively. Comparing with system (5), wehave

f (S, I ) = b + α − βSI − (b + α)S − αI, g(S, I ) = βSI − bI − γ I,

ξ(S, I ) = −mS, η(S, I ) = −pI , ϕ(S, I ) = I − H , (φ(T ),ψ(T )) = (φ(T ),H), and(φ(T +),ψ(T +)) = ((1 − m)φ(T ), (1 − p)H). Thus,

∂f

∂S= −βI − (b + α),

∂g

∂I= βS − b − γ, (9)



and

∂ξ

∂S= −m,

∂η

∂I= −p,

∂ϕ

∂I= 1,

∂ξ

∂I= ∂η

∂S= ∂ϕ

∂S= 0. (10)

Furthermore, it follows from (9) and (10) that

κ = (∂η∂I

∂ϕ∂S

− ∂η∂S

∂ϕ∂I

+ ∂ϕ∂S

)f+ + (∂ξ∂S

∂ϕ∂I

− ∂ξ∂I

∂ϕ∂S

+ ∂ϕ∂I

)g+∂ϕ∂S

f + ∂ϕ∂I

g

= (1 − m)g+(φ(T +),ψ(T +))

g(φ(T ),ψ(T ))

= (1 − m)(1 − p)[β(1 − m)φ(T ) − b − γ ]βφ(T ) − b − γ

(11)

and

μ = κ exp

{∫ T

0

[βφ(t) − b − γ − (

βψ(t) + b + α)]

dt

}

. (12)

On the other hand, we integrate the both sides of the second equation of system

(3) along the orbit E+E to give

ln1

1 − p=

∫ H

(1−p)H

dI

I=

∫ T

0[βS − b − γ ]dt =

∫ T

0

[βφ(t) − b − γ

]dt. (13)

From (11)–(13), we obtain

|μ| =∣∣∣∣(1 − m)(1 − p)[β(1 − m)φ(T ) − b − γ ]

βφ(T ) − b − γ

∣∣∣∣

1

1 − p

× exp

{

−∫ T

0

[βψ(t) + b + α

]dt

}

=∣∣∣∣(1 − m)[β(1 − m)φ(T ) − b − γ ]

βφ(T ) − b − γ

∣∣∣∣ exp

{

−∫ T

0

[βψ(t) + b + α

]dt

}

.

By condition (8), we see that system (3) satisfies all conditions of Lemma 2. Itthen follows from Lemma 2 that the order-1 periodic solution (φ,ψ) of system (3)is orbitally asymptotically stable and has asymptotic phase property. This completesthe proof. �

From the proof of Theorem 3, integrating both sides of the first equation of system

(3) along the orbit E+E, we obtain

ln1 − (1 − m)φ(T )

1 − φ(T )=

∫ φ(T )

(1−m)φ(T )

dS

1 − S≤

∫ T

0(b + α)dt = (b + α)T .



This shows that

exp

{

−∫ T

0

[βψ(t) + b + α

]dt

}

< exp{−(b + α)T

} ≤ 1 − φ(T )

1 − (1 − m)φ(T ).

The succeeding Corollary 1 is a direct consequence of Theorem 3.

Corollary 1 Let (φ,ψ) be a positive order-1 periodic solution of system (3) withperiodic T . If

|μ| =∣∣∣∣(1 − m)[β(1 − m)φ(T ) − b − γ ]

βφ(T ) − b − γ

∣∣∣∣

1 − φ(T )

1 − (1 − m)φ(T )< 1,

then (φ,ψ) is orbitally asymptotically stable.

Remark 4 From the numerical simulation in next section, we come to a conclu-sion that system (3) has a positive order-1 periodic (ϕ,ψ) for any m,p ∈ (0,1) andH ≤ I ∗, and (ϕ,ψ) is orbitally asymptotically stable. We thus raise a conjecture asfollows.

Conjecture 1 For any m,p ∈ (0, 1) and H ≤ I ∗, system (3) has a positive order-1periodic solution, which is orbitally asymptotically stable.

3.2 The Case of H > I ∗

Theorem 4 For any m,p ∈ (0, 1) and H > I ∗, one of the following statements isvalid.

(i) If there is a positive constant S = ρ(H) ∈ (0, S∗) such that the trajectoryO+(A0, t0) of system (3) starting from the initial point A0(S, (1 − p)H) is tan-gent to the line L : I = H at point B((b + β)/γ,H) and (1 − m)(1 − H) < S,then system (3) has a positive order-1 or order-2 periodic solution, which isorbitally asymptotically stable.

(ii) If for any S ∈ (0, S∗), the trajectory O+(A0, t0) of system (3) starting from theinitial point A0(S, (1 − p)H) cuts the line L : I = H at point B(S0,H), whereS0 > S∗, then system (3) has a positive order-1 or order-2 periodic solution,which is orbitally asymptotically stable.

(iii) If for any S ∈ (0, S∗), the trajectory O+(A0, t0) of system (3) starting fromthe initial point A0(S, (1 − p)H) does not intersect the line L : I = H , thensystem (3) has no positive order-k (k ≥ 1) periodic solution and the endemicequilibrium E∗(S∗, I ∗) is globally asymptotically stable.

Proof We first prove (i). If there is a positive constant S = ρ(H) ∈ (0, S∗) suchthat the trajectory O+(A0, t0) of system (3) starting from the initial point A0(S, (1 −p)H) crosses section Γp at point A0p(S1, (1−p)H) and is tangent to the line L : I =H at point B(S∗,H), then the trajectory starting from the initial point (S, (1 − p)H)



with S ∈ (S, S1) will tend to endemic equilibrium E∗(S∗, I ∗) and not intersect withsection ΓH . Moreover, if (1 − m)(1 − H) < S, then (1 − p)S < S for any point(S, I ) ∈ ΓH . Therefore, for any E(S,H) ∈ ΓH , trajectory O+(E, t0) starting fromthe initial point E(S,H) will intersect with ΓH infinitely many times due to theimpulsive effects �S(t) = −mS(t) and �I (t) = −pH .

On the other hand, for any two points Ei(Si,H) and Ej(Sj ,H) on section ΓH ,where Si , Sj ∈ (S∗, 1 − H) and Si < Sj , in view of impulsive effects, E+

i ((1 −m)Si, (1 − p)H) is on the left of E+

j ((1 − m)Sj , (1 − p)H). We claim that

0 < Sj+1 < Si+1 < 1 − H. (14)

In fact, if (14) is false, that is, Sj+1 ≥ Si+1, then Ej+1(Sj+1,H) is on the rightof Ei+1(Si+1,H), or the two points coincide. So we have trajectories O+(E+

i , t0)

and O+(E+j , t0) intersect at some point (S∗

0 , I ∗0 ). This implies that there are two

different solutions which start from the same initial point (S∗0 , I ∗

0 ), which contradictsthe uniqueness of solution. Inequality (14) is thus valid.

Suppose that the trajectory O+(E0, t0) of system (3) starting from the initial pointE0(S0,H) on section ΓH jumps to point E+

0 (S0, (1 −p)H) on section Γp due to im-pulsive effects �S(t) = −mS(t) and �I (t) = −pH and reaches section ΓH at pointE1(S1,H) again, where S1 ∈ (S∗, 1 −H), and then jumps to point E+

1 (S1, h) at sec-tion Γp . At state E+

1 , trajectory O+(E0, t0) intersects section ΓH at point E2(S2,H),where S2 ∈ (S∗, 1 − H). By the Poincaré map (4) of section ΓH , it follows thatS1 = F(S0,m,p,H) and S2 = F(S1,m,p,H). Repeating the above process, wehave Sn+1 = F(Sn,m,p,H) (n = 0,1, . . .). In particular, system (3) has a positiveorder-1 periodic solution when S0 = S1 and a positive order-2 periodic solution whenS0 �= S1 and S0 = S2.

Next, we look at the general situation where S0 �= S1 �= S2 �= · · · �= Sk (k > 2). Wediscuss the problem in different cases.

Case 1. S0 < S1. In this case, it follows from (14) that S2 < S1. This results in therelation of S0, S1, and S2 to be one of the following cases.

(1) S2 < S0 < S1. In this case, S3 > S1 > S2 by (14). Repeating the above process,we have

S∗ < · · · < S2k < · · · < S2 < S0 < S1 < · · · < S2k+1 < · · · < 1 − H.

(2) S0 < S2 < S1. Similar to 1), we have

S0 < S2 < · · · < S2k < · · · < S2k+1 < · · · < S3 < S1 < 1 − H.

Case 2. S0 > S1. In this case, it follows from (14) that S1 < S2. This results in therelation of S0, S1, and S2 to be one of the following cases.



(3) S1 < S0 < S2. In this case, S2 > S1 > S3 by (14). Repeating the above process,we have

S∗ < · · · < S2k+1 < · · · < S1 < S0 < S2 < · · · < S2k < · · · < 1 − H.

(4) S1 < S2 < S0. Similar to (3), we have

S∗ < S1 < · · · < S2k+1 < · · · < S2k < · · · < S2 < S0 < 1 − H.

Moreover, in (1) of Case 1, we have limk→∞ S2k = λ2 and limk→∞ S2k+1 = λ1,where S∗ < λ2 < λ1 < 1−h. Hence, λ1 = F(λ2,m,p,H) and λ2 = F(λ1,m,p,H).So system (3) has an orbit of asymptotically stable positive order-2 periodic solution.Similarly, in (2) of Case 1 and (4) of Case 2, system (3) has an orbital asymptoticallystable positive order-1 periodic solution. In (3) of Case 2, system (3) has an orbit ofasymptotically stable positive order-2 periodic solution. This proves (i).

Now we turn to (ii). If for any S ∈ (0, S∗), the trajectory O+(A0, t0) of system(3) starting from the initial point A0(S, (1 − p)H) cuts the line L : I = H at pointB(S0,H), where S0 > S∗, then for any E(S,H) ∈ ΓH , trajectory O+(E, t0) willintersect with section ΓH infinitely many times due to the impulsive effects �S(t) =−mS(t) and �I (t) = −pH . Similar to (i), we can also obtain that system (3) has apositive order-1 or order-2 periodic solution, which is orbitally asymptotically stable.(ii) thus follows.

Finally, we show (iii). If for any S ∈ (0, (b + β)/γ ), the trajectory O+(A0, t0)

of system (3) starting from the initial point A0(S, (1 − p)H) does not intersect theline L : I = H , then the trajectory starting from the point (S, (1 − p)H) of sectionΓp with S ∈ (0, (b + β)/γ ) will tend to endemic equilibrium E∗(S∗, I ∗) and notintersect with section ΓH . Furthermore, any other trajectory intersects section ΓH

at most finitely many times, and then tends to endemic equilibrium E∗(S∗, I ∗). Inthis case, system (3) has no positive order-k (k ≥ 1) periodic solution and endemicequilibrium E∗(S∗, I ∗) is globally asymptotically stable. This is (iii). The proof iscomplete. �

Remark 5 From part (i) of Theorem 4, we note that (1−m)(1−H) < S is a sufficientcondition for system (3) to has a positive order-1 or order-2 periodic solution.

Remark 6 Part (iii) of Theorem 4 shows that the state dependent pulse effects areinvalid when the vaccination m and medication intensification effort p remain at arelatively low level and threshold value H is greater than I ∗.

4 Numerical Simulation and Discussion

To illustrate the results and the feasibility of the state dependent pulse feedback con-trol strategy, we consider the following SIRS epidemic system with the state depen-



Fig. 2 The orbitally asymptotically stable period solution of system (15) with m = 0.3, p = 0.6, andH = 0.3 < I∗ (Color figure online)

dent pulse vaccination and medication:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dS(t)

dt= 0.15(S(t) + I (t) + R(t)) − 0.8S(t)I (t) − 0.15S(t) + 0.1R(t),

dI (t)

dt= 0.8S(t)I (t) − 0.15I (t) − 0.2I (t),

dR(t)

dt= 0.2I (t) − 0.15R(t) − 0.1R(t),

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

I < H,

�S(t) = S(t+

) − S(t) = −mS(t),

�I (t) = I(t+

) − I (t) = −pH,

�R(t) = R(t+

) − R(t) = pH + mS(t),

⎫⎪⎪⎬

⎪⎪⎭

I = H.

(15)It is obvious that system (15) without the pulse effects has a unique globally asymp-totically stable endemic equilibrium (S∗, I ∗,R∗) = (0.4375,0.3125,0.25).

Firstly, we choose the control parameters to be m = 0.3, p = 0.6, and H = 0.3 <

I ∗ = 0.3125, respectively. By Theorem 2, we know that system (15) has a positiveorder-1 periodic solution (φ,ψ,ϕ), which is shown in Figs. 2(a)–(c). At the sametime, simulation results also show that the periodic solution starting from initial point(0.4753,0.06,0.4647).



Fig. 3 The trajectory of system (15) with p = 0.6, H = 0.3242 > I∗ = 0.3125, and m = 0.85 in (a) andm = 0.55 in (b), respectively (Color figure online)

In addition, it is easily to calculate that

|μ| =∣∣∣∣(1 − m)[β(1 − m)φ(T ) − b − γ ]

βφ(T ) − b − γ

∣∣∣∣

1 − φ(T )

1 − (1 − m)φ(T )

≈ 0.7(0.8 × 0.4753 − 0.15 − 0.2)

0.8 × 0.679 − 0.15 − 0.2× 1 − 0.679

1 − 0.4753

≈ 0.067 < 1.

Therefore, the order-1 periodic solution of system (15) is orbitally asymptotically sta-ble and has asymptotic phase property by Corollary 1, which is shown in Fig. 2(d). Inaddition, further more numerical simulations show that for any m,p ∈ (0,1), system(15) with H < I ∗ has a positive orbitally asymptotically stable order-1 periodic solu-tion. This is just Conjecture 1 in Sect. 3. This shows that the conditions of Theorem3 or Corollary 1 are sufficient not necessary.

Next, let m = 0.9, p = 0.6, H = 0.3242, and (1 − m)(1 − H) = 0.06758 < S =0.098. It is easy to see that system (15) has a positive orbitally asymptotically stableorder-1 periodic solution by part (i) of Theorem 4, which is shown in Fig. 3(a). How-ever, if we choose m = 0.55, p = 0.6, and H = 0.3242, numerical simulations showthat the trajectories of system (15) intersect the line L : I = H at most finitely manytimes and then tend to endemic equilibrium (0.4375,0.3125,0.25), which is shownin Fig. 3(b). This is just part (iii) of Theorem 4. This implies that system (15) has nopositive order-k (k ≥ 1) periodic solution in this case.

Thirdly, we discuss how the state dependent pulse control strategy affects the pre-vention and control of infectious diseases and the existence and stability of the peri-odic solutions. Here, we choose H = 0.1 < I ∗, p = 0.6, and m to be 0.4, 0.6, and0.8, respectively. Numerical simulations show that the period T of order-1 periodicsolution for system (15) increases with the increase of immune strength m, which isshown in Fig. 4(a). Furthermore, let H = 0.1, m = 0.55, and p to be 0.4, 0.6, and 0.8,respectively. The corresponding numerical results are presented in Fig. 4(b). Theseresults indicate that the recurrence time for infectious diseases can be prolonged byincreasing immune strength m or medication intensification effort p. That is, the in-fected density can be kept at a low level by adjusting immune or medication strength



Fig. 4 The trajectory of system (15) with H = 0.1 and (a): p = 0.6, m = 0.4,0.6,0.8, respectively;(b) m = 0.55, p = 0.4,0.6,0.8, respectively (Color figure online)

Fig. 5 The trajectory of system (15) with H = 0.32 > I∗ and (a): p = 0.4, m = 0.2,0.4,0.6,0.8,0.9,respectively; (b) m = 0.6, p = 0.2,0.4,0.6,0.8,0.9, respectively (Color figure online)

(see Fig. 6(a)). The strong consistency between theoretical result and real situation isobviously observed.

However, if H = 0.32 > I ∗, p = 0.4, and m to be taken 0.2, 0.4, 0.6, 0.8, and 0.9,respectively, then unique endemic equilibrium (S∗, I ∗,R∗) of system (15) is asymp-totically stable when the immune strength m remains at a relatively low level. How-ever, with the increase of immune strength m, (S∗, I ∗,R∗) loses its global asymptoticstability and system (15) has a positive order-1 orbitally asymptotically stable peri-odic solution. This is observed in Fig. 5(a). Similar results can also be obtained fromH = 0.32 > I ∗, m = 0.6, and p to be taken as 0.2, 0.4, 0.6, 0.8, and 0.9, respectively.This is presented in Fig. 5(b). Numerical simulations also indicate that we can con-trol infectious disease at a relatively low level over a long period of time by adjustingimmune or medication strength when H < I ∗ or H > I ∗ (see Figs. 6(a) and (b)).

Finally, we compare the state dependent pulse control strategy and FTPS. First, wegive a cost of control measures. Assume that the control cost is proportional to thedensities of treated infected group and immunized susceptible group. For simplicity,we use the following total cost:

C =∑

mS(ti (H)

) + pI(ti (H)

). (16)



Fig. 6 The trajectory of system (15) with (a): H = 0.2 < I∗, (b): H = 0.32 > I∗ (Color figure online)

Fig. 7 The state dependent pulse vaccination and medication vs. the fixed time impulsive vaccination andmedication, where m = 0.35, p = 0.4, and (a) I (0) = 0.04, (b) I (0) = 0.02 and 0.04, respectively (Colorfigure online)

The disease is controlled by the impulsive effects �S(t) = −0.35S(t), �I (t) =−0.4H , and �R(t) = 0.4H + 0.35S(t). Numerical simulations show that infectedgroup is controlled within a relatively low level by the state dependent pulse con-trol when the number of infected population reaches the hazardous threshold. Thisis well explained in Fig. 7. Assume that the hazardous threshold is 0.05 and ini-tial value is (0.6,0.04,0.36). The trajectory of this solution is (0.6,0.04,0.36) →(0.6825,0.05,0.2675) → (0.7226,0.05,0.1774) → (0.7232,0.05,0.2268) →(0.7232,0.05,0.2268) → ·· · . By (16), we have the control cost as follows:

C1 = 0.35 × 0.6 + 0.4 × 0.04 = 0.226.

When the infected group is controlled within 0.05 at t ≈ 1.346, the solution tendsquickly to a stable periodic solution by state feedback control strategy, in otherwords, the disease is completely controlled. However, for the same initial value(0.6,0.04,0.36), if we take control measures at a fixed time t = 4k (k = 1,2, . . .), thedensity of infected population can not controlled under the hazardous threshold 0.05,and hence the disease will be spreading. Furthermore, if we take control for the in-fected population at a fixed time T = 3k (k = 1,2, . . .), the trajectory of this solutionwith initial value (0.6,0.04,0.36) is (0.6,0.04,0.36) → (0.7347,0.0719,0.1934) →



(0.6878,0.0639,0.2483) → (0.6834,0.0545,0.2621) → (0.69,0.0467,0.2633) →·· · . The infected population is controlled within 0.05 at t ≈ 11.998 and the controlcost is

C2 = 0.35(0.6 + 0.7347 + 0.6878 + 0.6834) + 0.4(0.04 + 0.0719

+ 0.0639 + 0.0545) ≈ 1.039.

Therefore, if we choose the same immune and medication strength m and p andthe appropriate control parameters, the state dependent pulse control would be veryeconomic.

In Fig. 7(a), we also note that the fixed time control with T = 3k is slightly fre-quent than the state dependent control. However, if the initial value of infected popu-lation is changed, FTPS is difficult to achieve the expected purpose, which is observedin Fig. 7(b). We can therefore say that the state dependent pulse vaccination and med-ication is more effective and easier implementable than the fixed time pulse controlstrategy.

5 Concluding Remarks

The dynamic behavior of a SIRS epidemic model with state dependent pulse controlstrategy is studied in this paper. The state dependent pulse control strategy causesthe complexity for the dynamic behavior of system (2) such as frequent switchingbetween states, irregular motion, and some uncertainties. This is the distinguishedfeature compared with other control strategies.

By the Poincaré map, the analogue of Poincaré criterion, and qualitative analysismethod, some sufficient conditions on the existence and orbital stability of positiveorder-1 or order-2 periodic solution of system (2) are presented. This amounts to thatwe can control the density of infectious disease at a low level over a long periodof time by adjusting immune or medication strength. It is concluded that the statedependent pulse control strategy is more feasible, effective, and importantly, easierimplementable than the fixed-time pulse control.

Finally, we mention some future possible works along the present work: (a) theuniqueness and global stability of positive order-1 periodic solution for H < I ∗;(b) the optimal control strategy for state dependent pulse control problem; (c) ap-plication to more complicated epidemic models.

Acknowledgements The authors would like to thank antonymous referees for their constructive sug-gestions and comments that improve substantially the original manuscript.

This work was supported in part by the Natural Science Foundation of Xinjiang (Grant No.2011211B08), the National Natural Science Foundation of China (Grant No. 11001235, 11271312, and11261056), the China Postdoctoral Science Foundation (Grant No. 20110491750 and 2012T50836), theScientific Research Programmes of Colleges in Xinjiang (Grant No. XJEDU2011S08), the National BasicResearch Program of China (2011CB808002), and the National Research Foundation of South Africa.

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Documents

lsc.amss.ac.cnlsc.amss.ac.cn/~bzguo/papers/Nielf.pdf · 2013-10-07 · 1698 L.-F. Nie et al. (1927) in the late 1920s. In this model, the total population N is divided into sus- ceptible